Discounts, sales tax and tips

An 80-dollar jacket at 25% off costs 60 dollars, and you can get there in one step by working out 75% of 80.

20% off, then another 10% off. Is that 30%?

Try this before reading on. A shop marks a coat 20% off, and at the till it takes another 10% off the reduced price. Is that 30% off altogether? Write down an answer and one reason. The tools to check it come first, and the answer comes near the end.

Finding 25% of 80

Almost every shop calculation is one sentence: find a percent of an amount. Change the percent to a decimal and multiply. 25% of 80 is 0.25 × 80 = 20. OpenStax writes the three everyday versions as formulas: the amount of discount is the discount rate times the original price, the sales tax is the tax rate times the purchase price, and a commission is the rate of commission times the sales.

The 80-dollar jacket at 25% off

A jacket costs 80 dollars and is marked 25% off. The discount is 0.25 × 80 = 20 dollars, and the sale price is the original price minus the discount: 80 − 20 = 60 dollars. There is a shorter road. If 25% comes off, you pay the other 75%, so the sale price is 0.75 × 80 = 60 in one multiplication. OpenStax puts this as a check question: the sale price should always be less than the original price.

Working backwards is common too. A coat is on sale for 51 dollars after 15% off. You paid 85% of the original, so the original was 51 ÷ 0.85 = 60 dollars. Dividing undoes the multiplication; subtracting 15% of 51 from 51 would not.

Sales tax added at the till

In many parts of the United States, as OpenStax describes, sales tax is not included in the shelf price; it is added when you pay. With a tax rate of 8%, a 60-dollar purchase has 0.08 × 60 = 4.80 dollars of tax, and the total cost is 64.80 dollars. Again one multiplication does it: you pay 100% + 8% = 108% of the price, and 1.08 × 60 = 64.80. Where a shelf price already includes tax, the shop has done this step for you.

Korea's 10% VAT, added and taken back out

In South Korea the value-added tax rate is 10%, set by Article 30 of the VAT Act, and the National Tax Service lists 10% for every type of business taxed at the general rate. A made-up meal with a supply price of 10,000 won costs 1.1 × 10,000 = 11,000 won with VAT. Going back the other way is where people slip. To find the price before tax inside 11,000 won, divide by 1.1: 11,000 ÷ 1.1 = 10,000. Taking 10% of 11,000 off gives 9,900, which is wrong, because the 10% was worked out on 10,000, not on 11,000.

Does it matter whether the discount or the tax comes first? Not for a percent discount. 80 × 0.75 × 1.08 and 80 × 1.08 × 0.75 are the same product, 64.80, because the order of multiplication does not change the answer.

An 18% tip on a 46-dollar meal

A tip is a percent of the bill chosen by the customer. For a 46-dollar meal and an 18% tip, the tip is 0.18 × 46 = 8.28 dollars, so the total is 54.28 dollars, or 1.18 × 46. To do it in your head, find 10% by moving the point one place: 10% of 46 is 4.60. Then 20% is twice that, 9.20, and 15% is 10% plus half of 10%, 4.60 + 2.30 = 6.90.

Back to the coat: 20% and then 10% is 28% off

A shop takes 20% off and then another 10% off the reduced price. That is not 30% off. You pay 0.8 × 0.9 = 0.72 of the price, which is 28% off. With a fixed-amount coupon the order does matter: on a 100-dollar item, taking 10 dollars off first and then 20% gives 90 × 0.8 = 72, while 20% first and then 10 dollars gives 80 − 10 = 70. OpenStax sets a coupon problem of exactly this kind.

1+1 and 2+1 as percents

Korean convenience stores run "1+1" offers (buy one, get one free) and "2+1" offers. 1+1 means two items for the price of one, so each costs half: 50% off. 2+1 means three for the price of two. You pay 2/3 of the full price per item, so the discount is 1/3, about 33.3%, not 50%. A made-up drink at 1,500 won on 2+1: three for 3,000 won, 1,000 won each.